Approximation Algorithms for Orienteering with Time Windows

Orienteering is the following optimization problem: given an edge-weighted graph (directed or undirected), two nodes s, t and a time limit T , find an s-t walk of total length at most T that maximizes the number of distinct nodes visited by the walk. One obtains a generalization, namely orienteering with time-windows (also referred to as TSP with time-windows), if each node v has a specified time-window [R(v), D(v)] and a node v is counted as visited by the walk only if v is visited during its time-window. For the time-window problem, an O(logOPT) approximation can be achieved even for directed graphs if the algorithm is allowed quasi-polynomial time. However, the best known polynomial time approximation ratios are O(log OPT) for undirected graphs and O(log OPT) in directed graphs. In this paper we make some progress towards closing this discrepancy, and in the process obtain improved approximation ratios in several natural settings. Let L(v) = D(v) − R(v) denote the length of the time-window for v and let Lmax = maxv L(v) and Lmin = minv L(v). Our results are given below with α denoting the known approximation ratio for orienteering (without time-windows). Currently α = (2 + ) for undirected graphs and α = O(log OPT) in directed graphs. • An O(α logLmax) approximation when R(v) and D(v) are integer valued for each v. • An O(αmax{logOPT, log Lmax Lmin }) approximation. • An O(α log Lmax Lmin ) approximation when no start and end points are specified. In particular, if Lmax Lmin is poly-bounded, we obtain an O(log n) approximation for the time-window problem in undirected graphs.